How Does Back-Reaction at the Conducting Glass Substrate Influence

The influence of the back-reaction on open circuit photovoltage decay measurements and on intensity modulated photovoltage (IMVS) measurements has bee...
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J. Phys. Chem. B 2005, 109, 7392-7398

How Does Back-Reaction at the Conducting Glass Substrate Influence the Dynamic Photovoltage Response of Nanocrystalline Dye-Sensitized Solar Cells? P. J. Cameron and L. M. Peter* Department of Chemistry, UniVersity of Bath, Bath BA2 7AY, United Kingdom ReceiVed: NoVember 19, 2004; In Final Form: February 7, 2005

In dye-sensitized nanocrystalline solar cells (DSC), the transfer of electrons from the conducting glass substrate to triiodide ions in solution is an important loss mechanism that can be suppressed by using thin compact blocking layers of TiO2. Whereas back-reaction at the substrate is relatively unimportant under short circuit conditions, it must be taken into account at the maximum power point or at open circuit. The influence of the back-reaction on open circuit photovoltage decay measurements and on intensity modulated photovoltage (IMVS) measurements has been studied by model simulations and by experimental measurements. The simulations demonstrate that reliable information about DSC properties such as trapping distributions can only be derived from transient or periodic photovoltage responses if the back-reaction is suppressed by the use of suitable blocking layers.

Introduction Interest in dye-sensitized nanocrystalline solar cells (DSC) has grown continually since the original Nature paper by O’Regan and Gra¨tzel.1 One of the remarkable features of the cell is that electrons injected into the porous film of nanocrystalline TiO2 are collected efficiently at the conducting glass substrate under short circuit conditions. This is surprising, because electrons appear to move through the oxide layer by a random walk process that can be modeled by multiple trapping with a distribution of trap energies.2,3 The porous titanium dioxide layer is permeated by the triiodide/iodide redox electrolyte, so that electrons can be transferred to I3- in a backreaction that represents a loss mechanism. Fortunately, this electron transfer process is slow, so that it can be neglected under short circuit conditions where the average transit time for the collection of electrons is shorter than the characteristic time constant for the back-reaction.4 Under open circuit conditions, by contrast, the back-reaction is the dominant route for establishing the photostationary state in which the rate of photoinjection from the dye is balanced by electron transfer to I3- and to the oxidized dye. Theoretical analysis of dynamic photovoltage experiments with DSC conducted under open circuit conditions is generally based on choosing a boundary condition corresponding to zero flux of electrons at the conducting glass substrate.5,6 This condition implies that the photoinjection of electrons is balanced by their back-reaction with I3- and with oxidized dye molecules at the TiO2/electrolyte interface. However, our recent work7 has shown that the possibility of electron transfer via the fluorinedoped tin oxide (FTO) coated glass substrate must be considered when formulating the boundary condition. At open circuit, the electron flux in the external circuit is zero, so that photoinjection must balance with the total rate of back-reaction via the nanocrystalline substrate and Via the substrate. The rate of electron transfer to I3- at the highly doped, and therefore quasimetallic, FTO substrate is potential dependent, increasing exponentially with the photovoltage. As a consequence, the * Corresponding author. E-mail: [email protected].

intensity dependence of the photovoltage can be distorted by back-reaction at the substrate, leading to substantial deviations from ideal diode-like behavior.7 This back-reaction can be largely eliminated if a thin blocking layer of TiO2 is deposited by spray pyrolysis,8 and cells with such a layer can exhibit an almost ideal diode-like dependence of the photovoltage on intensity.7 In the present contribution, we examine how back-reaction at the FTO influences the analysis of the photovoltage decay after interruption of continuous illumination.9-12 Consideration is also given to the small signal photovoltage response of the DSC to intensity-modulated illumination (intensity modulated photovoltage spectroscopy: IMVS), which provides essentially the same information. Experimental results are presented for DSC constructed with and without blocking layers, and it is shown that only the data obtained for cells with a blocking layer give reliable information about the properties of the nanocrystalline oxide, in particular the energetic distribution of electron traps. Theory Under open circuit conditions, the balance of photon and electron fluxes in the DSC can be expressed as

qI0 × IPCE ) jTiO2 + jsub

(1)

Here I0 is the incident photon flux density, IPCE is the incident photon to current conversion efficiency, jTiO2 is the current density associated with back-reaction via the conduction band and surface states in the nanocrystalline TiO2 and jsub is the current density for back-reaction via the FTO substrate.7 In a recent paper, Bisquert and Vihrenko12 have defined the quasi-static condition for DSC as one in which free and trapped electrons remain essentially in a common equilibrium, even when the system is displaced away from equilibrium by some perturbation (it is possible to solve the continuity equations numerically without making the quasi-stationary assumption: this point will be discussed elsewhere13). The quasi-stationary condition considerably simplifies the description of the system

10.1021/jp0407270 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/19/2005

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response. If the condition is fulfilled and back-reaction of electrons with I3- occurs only via the conduction band and not via surface states, it is possible to derive an analytical expression for the effective electron lifetime in terms of the density and energetic distribution of trapping states, st(E), which in the case of an exponential trap distribution can be defined for the interval EF,redox to Ec as

[

st(E) ) st(0) exp

]

β(E - EF,redox) kBT

(2)

Here st(0) is the value of the density of states function at the dark Fermi level, EF,redox, Ec is the energy of the conduction band and β is a parameter describing the width of the exponential trap distribution. It is, of course, possible to consider back-reaction of electrons via surface states in addition to the conduction band route. However, in an earlier paper,7 we presented experimental evidence that suggests that this route is relatively unimportant in the cells that we have studied, despite the high density of trap states. We concluded that the majority of traps are in the bulk of the TiO2 nanocrystals, so that electrons can only reach the surface via the conduction band. At open circuit, the relaxation of the density of electrons in the conduction band following a small increment or decrement of light intensity involves establishing a new photostationary state by adjusting the rate of the back-reactions and the occupancy of electron traps. Under illumination, the steady state density of conduction band electrons and the trap occupancy are both determined by the position of the quasi-Fermi level, nEF ) EF,redox + qUphoto, where Uphoto is the photovoltage. In most previous work, the time constant, τn, for this relaxation process has been referred to as the electron lifetime. This time constant should not be confused with the lifetime of free electrons in the conduction band determined by back-reaction alone. In the case where the back-reaction of electrons with oxidized species in solution is first order in electron concentration, we follow the approach of Bisquert and Vihrenko12 by defining a conduction band electron lifetime τn,0 in terms of a pseudo-first-order rate constant kcb:

τn,0-1 ) k′cb[O] ) kcb

(3)

It should be noted that this lifetime is generally much shorter than the observable relaxation time constant for electrons in the conduction band, because the latter is influenced by the time required to establish the photostationary distribution of electrons in traps. Bisquert and Vihrenko12 have shown that under quasi-static conditions, the characteristic time constant τn for relaxation of the conduction band electron density (and hence of the photovoltage) is related to β by the power law expression (note that eq 32 in ref 12 has a typographical error)

Nt τn ) β βnβ-1 c τn,0 Nc

(4)

Here Nc is the effective density of states in the conduction band, Nt is the total density of trap states, and nc is the density of electrons in the conduction band. Equation 4 can be rewritten by noting that nc is related to the photovoltage by

nc ) nc,0 exp

( ) qUphoto kBT

(5)

where nc,0 is the density of conduction band electrons in the dark, given by

(

)

Ec - EF,redox kT

nc,0 ) Nc exp -

(6)

It follows that a semilogarithmic plot of the time constant τn as a function of photovoltage should fit the expression

qUphoto log10 τn ) const + (β - 1) 2.303kBT

(7)

and therefore have a slope of (β - 1)/2.303kBT. Zaban et al.11 have discussed the determination of τn from the decay of the open circuit photovoltage of DSC following interruption of illumination, and this treatment has been reviewed by Bisquert and Vihrenko.12 Under quasi-static conditions, τn can be obtained from photovoltage decay curves by noting that

τn ) -

(

)

kBT dUphoto q dt

-1

(8)

Alternatively, IMVS4,14 can be used, and in this case τn is found from the frequency (fmin) of the minimum in the modulated photovoltage (τn ) 1/2π fmin). In previous work,9 we have measured the rate of decay of the total (i.e., free + trapped) electron density in the DSC at open circuit. This was done by short circuiting the cell after a predetermined delay during the photovoltage decay and integrating the resulting current transient to determine the residual electron concentration. It follows from the preceding treatment that the “electron lifetime”, τn,tot measured by this charge extraction technique is not identical with the time constant derived from photovoltage decay measurements or IMVS. For the case where the trap distribution is exponential, it can be shown that the ratio of the two different time constants is simply equal to β

τn )β τn,tot

(9)

The preceding derivation assumes that the only route for the back-reaction of electrons with I3- is via the conduction band of the nanocrystalline TiO2. In principle, electrons can also be transferred via surface states, and in our previous paper on this topic,7 we used a global rate constant to describe this process. Recently, Bisquert et al.15 have given a more detailed treatment that uses the Marcus Gerischer approach to model charge transfer from surface states to redox species in solution. However, the assumption that back-reaction can occur only via the conduction band and surface states is not justified unless back-reaction via the substrate can be neglected. Generally this is not the case unless a blocking layer is used. The time constant τn in the case where back-reaction via the substrate is important can be derived numerically using the flux balance expression in eq 1. The procedure involves definition of the current densities associated with back-reaction via the nanocrystalline oxide and via the substrate in terms of the photovoltage:

jcb ) -qdkcbnc,0 exp

( ) qUphoto kBT

(10)

7394 J. Phys. Chem. B, Vol. 109, No. 15, 2005

[

jsub ) -j0 exp

Cameron and Peter

]

-(1 - R)nqUphoto RnqUphoto - exp kBT kBT

(11)

The second term corresponds to the familiar Butler-Volmer equation for the electron transfer process at a metal electrode. j0, the exchange current density, is a measure of the ease with which electrons are exchanged with the redox system. R, the cathodic transfer coefficient, determines how the photovoltage affects the current (note that the photovoltage corresponds to the overpotential in a conventional electrochemical experiment). n is the number of electrons transferred in the rate determining step. Under the conditions where the electron density in the conduction band is much smaller than the total density of trapped electrons, τn is given in the quasi-static approximation by

τn )

( )

∂nt τ ∂nc n,0

(12)

Here nt is the density of trapped electrons, which is given by the integral

nt )

∫EE

F,redox+qUphoto

F,redox

ft(E) st(E) dE

(13)

where ft(E) is the Fermi-Dirac occupancy function. nt and nc can be defined in terms of the photovoltage, and the rate of change of total electron density (and hence charge) is defined by the sum jTiO2 + jsub. The photovoltage decay curve can therefore be simulated for given values of j0 (back-reaction via the substrate) and τn,0-1 (back-reaction via the conduction band of the nanocrystalline TiO2). In general, the density of states function st(E) can be obtained from the measured values of τn as a function of Uphoto using the following relationship (note that in the case of an exponential distribution of trapping states, the β term is included via τn (cf. eq 4)]:

st(E) )

( )( ) ( ) ( ) (

τn nc,0 qUphoto exp ) τn,0 kBT kBT τn Nc - (Ec - EF + qUphoto) (14) exp τn,0 kBT kBT

)

Experimental photovoltage decay curves for thin films can also be influenced by the excess charge, Qsub, stored in the double layer of the FTO (or in the space charge layer of a blocking layer). This charge can be approximated by

Qsub = CsubUphoto

(15)

if it is assumed that the double layer capacitance, Csub, of the substrate is constant. Unless the nanocrystalline film is thin, this effect can be neglected to a first approximation. However, it was incorporated into the simulation program to ensure generality. Simulation Results Simulations of the photovoltage decay were carried out using an exponential distribution of trapping states with β ) 0.25 and different values of the exchange current density j0 (see Appendix for values used in the calculations: experimental values of j0 for different conducting glass substrates are given in ref 7). Although the simulation model allows for the possibility of reaction via surface states, to simplify the diagnostic criteria, it

Figure 1. (a) Comparison of photovoltage decay plot simulated for the case where no back-reaction takes place via the substrate with plots for cases where the back-reaction is important. j0 is the exchange current density for the back-reaction. (b) Plot of simulated photovoltage decay vs log t showing the linear behavior predicted for the case where the trap density of states function is exponential and back-reaction is negligible. By contrast, in the case where back-reaction takes place at the substrate, the semilogarithmic plot is nonlinear.

was assumed that back-reaction in the nanocrystalline film occurs entirely via the conduction band. Figure 1a illustrates the predicted influence of back-reaction via the substrate on the open circuit photovoltage decay measured after interruption of illumination. It is clear that even very low values of j0 lead to a significant effect on the photovoltage decay at longer times. Figure 1b shows that a plot of Uphoto vs log t for the ideal case (j0 ) 0) is linear at times greater than 0.1 s, whereas the corresponding decay plots for j0 * 0 are nonlinear. Figure 2 shows that back-reaction via the substrate becomes the dominant process at longer times because it varies less rapidly with photovoltage (Rn ) 0.5 in eq 11) than reaction via the conduction band (cf. eqs 9 and 10). It follows that derivation of the time constant τn from photovoltage decay plots at longer times is subject to significant errors unless backreaction via the substrate is prevented by using a blocking layer. Figure 3 contrasts the dependence of τn on photovoltage that is expected if back-reaction via the substrate is negligible with the dependence calculated for the case where back-reaction via the substrate is included. The simulation was performed for a β value of 0.25, so in the ideal case we expect the slope of the semilogarithmic plot to be 79 mV/decade (cf. eq 7 with β ) 0.25). The figure confirms that this is indeed the case provided that j0 ) 0. However, when back-reaction via the substrate is

Photovoltage Response of Nanocrystalline Solar Cells

Figure 2. Plot showing how the decay route changes during the photovoltage transient in the case where back-reaction a takes place at the substrate. At high photovoltages, back-reaction via the conduction band dominates, but back-reaction via the conducting glass substrate becomes more important as the photovoltage decreases. Note that the calculation has been performed for a very low value of exchange current density j0, but even here the effect is substantial.

Figure 3. Plots of relaxation time constant τn vs photovoltage derived from simulated photovoltage decay transients for different values of j0. Note that the effect of back-reaction at the substrate is to reduce τn by providing an additional route for electron decay. The results show that transients for j0 * 0 only give reliable information about the trap distribution s(E) at high photovoltages, whereas at lower photovoltages the back-reaction distorts the plots. At lower photovoltages, the slope of the plot is determined by the value of Rn (cf. eq 11).

included in the simulation, the lifetime plots deviate substantially from ideal behavior a soon as the photovoltage decays from its initial value. This can be rationalized by recognizing that backreaction via the substrate decreases the time constant τn by adding an additional relaxation pathway. As a consequence, the dependence of τn on photovoltage becomes determined in part by the Butler-Volmer equation, and the slope of the semilogarithmic plot decreases, in this case toward a limiting slope of 226 mV/decade. If this limiting slope were (mis)interpreted to obtain β, an entirely incorrect value of β ) 0.76 would be obtained. In our earlier work,6,16 τn was determined by IMVS as a function of intensity, and the data were presented as double logarithmic plots of τn vs photon flux. The simulated plots in Figure 4 show how the back-reaction at the substrate affects this kind of plot. In the absence of back-reaction via the

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Figure 4. Plots of τn as a function of absorbed photon flux density calculated for different values of the exchange current density j0. It can be seen that back-reaction at the substrate causes significant deviations in the slope of the plot from the ideal value of 1 - β (in this case 0.75).

Figure 5. Plots of the apparent density of states function derived from simulated photovoltage decay transients. It can be seen that even a very low value of j0 suffices to give an entirely false value of the broadening factor β. It follows that st(E) is only accessible from experimental data if measurements are made on DSC with good quality blocking layers.

substrate, the plot is linear with a slope of β - 1 (in this case, β ) 0.25, so the slope is -0.75). However, even a low value of j0 leads to significant deviation from ideal behavior, and the slope of the plot decreases. If back-reaction via the substrate becomes dominant, the slope tends toward Rn (0.5 in this calculation) because it is given by the Tafel limit of the ButlerVolmer equation. In previous work,6 a slope close to 0.5 in the double logarithmic plot was interpreted as evidence that the back-reaction of electrons with I3- is second order in electron concentration as a consequence of a two-step mechanism. In view of the evident influence of back-reaction via the substrate, this conclusion is no longer valid. All of these simulations suggest that extreme caution must be exercised when interpreting photovoltage decay and IMVS data. This is particularly true when it comes to attempts to derive the density of states function for electron traps. Figure 5 illustrates the apparent st(E) values derived using eq 14. The ideal behavior occurs when j0 ) 0, when the slope of the semilogarithmic plot is equal to β (cf. eq 2). By contrast, even a very low value of j0 (10-10 A cm-2) suffices to distort the

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Cameron and Peter

plot completely so that it is no longer possible to derive the correct value of β. It follows that the DOS function for electron traps can only be derived using cells with exceptionally good blocking layers. Experimental Section Dye sensitized solar cells were prepared with and without TiO2 blocking underlayers. TiO2 blocking layers17 were prepared as described previously8 by spraying a 0.2 M solution of titanium di-isopropoxide bis(acetylacetonate) in 2-propanol onto the hot conducting glass substrate (450 °C) in 26 short bursts over two minutes. The aqueous nanocrystalline TiO2 colloid was made by the acetic acid peptization route described elsewhere.4,5 The TiO2 films, which were typically 10 µm thick, were sensitized with cis-bis(isothiocyanato) bis(2,2′-bipyridyl-4,4′-dicarboxylato)ruthenium(II) bis(tetrabutylammonium) (N719). The redox electrolyte, consisted of 0.85 M methylhexylimidazolium iodide, 0.1 M LiI (anhydrous 99.999%, Aldrich), 0.05 M I2 (anhydrous 99.999%, Aldrich) and 0.2 M tert-butylpyridine (99%, Aldrich) in acetonitrile. Photovoltage decay and IMVS measurements were made using systems described previously.6,18 The excitation wavelength was 470 nm. Results and Discussion Parts a and b of Figure 6 show how incorporation of a thin blocking layer of TiO2 influences the photovoltage decay in the case of a fluorine doped tin oxide (FTO) coated substrate (Libby Owens Ford TEC 15 supplied by Hartford) that is known to be active for the reduction of I3- when it has been thoroughly cleaned.7 In the absence of the blocking layer, the photovoltage decays almost entirely within 10 s of the interruption of illumination. By contrast, the decay is much slower when a blocking layer is present. This shows that the dominant route for the reaction of electrons with I3- is via the FTO in the first case (cf. Figure 1): our previous work7 has shown that j0 can be as high as 10-8 A cm-2 for these substrates. In the second case, the contribution of electron transfer via the FTO is relatively minor, and the dominant route involves electron transfer to I3- from the nanocrystalline TiO2 layer. For cells constructed without blocking layers, the rate of photovoltage decay depends on the type of FTO used and on the extent to which it has been cleaned. By contrast, cells with blocking layers give reproducible results. Contamination of the FTO surface may be beneficial to cell performance if it lowers the exchange current density for the I3-/I- couple, and this may explain some of the variability in the properties of conventional DSC without blocking layers. In this context, we note that Zhu et al.19 concluded that the locus of recombination was close to the substrate in series of cells that were studied using a modulated infrared absorption technique developed in our laboratory.20 The relaxation time constant τn can be derived from the photovoltage decay using eq 8. Figure 6b shows that in the case of the film with a blocking layer, a plot of photovoltage vs log t is linear, as expected when the density of surface states function st(E) is exponential (compare Figure 1b). Numerical differentiation of the smoothed decay plot was used to obtain the derivative (dUphoto/dt) to calculate τn as a function of photovoltage (cf. eq 8). The same procedure was used to for the cell without a blocking layer, to derive τn. Figure 7 compares the results of the analysis obtained in the case of transients shown in Figure 6 for cells with and without blocking layers. It can be seen that the relaxation time constant of the cell with a blocking layer appears to fit quite well to the behavior expected for an exponential density of states distribu-

Figure 6. (a) Experimental photovoltage decay plots for cells with and without TiO2 blocking layers. Note the extremely rapid decay of the photovoltage for the cell with a clean FTO substrate without a blocking layer. (b) Semilogarithmic plots of the transients in (a), showing the linear behavior characteristic of a cell with blocking layer and the nonlinear behavior of the cell without blocking layer.

tion: the slope of 71 mV/decade corresponds to β ) 0.19 (cf. eq 7). By contrast, the cell without a blocking layer gives much lower values of τn over the entire photovoltage range, indicating that the electron density decays in part via the FTO substrate. It is worth noting that the linear part of the plot agrees well with the simulations shown in Figure 3. Clearly, it is not possible to derive meaningful information about trap distributions by analyzing the decay of the photovoltage for the DSC without a blocking layer. Figure 7 also contrasts the values of τn obtained by analysis of the photovoltage decay for the two cells with those obtained by IMVS. In the case of the cell with the blocking layer, the τn values obtained by IMVS coincide with those obtained by analysis of the photovoltage transient. In the case of the cell without a blocking layer, the IMVS data lie above the line calculated from the photovoltage decay. This may indicate that

Photovoltage Response of Nanocrystalline Solar Cells

Figure 7. Comparison of experimental dependence of τn on Uphoto for DSC with and without blocking layers. Note the excellent agreement between the values of τn derived from IMVS and the photovoltage decay transient in the case of the cell with a blocking layer. The slope of the line shown is 71 mV/decade, which corresponds to a β value of 0.19 (cf. eq 7). In the case of the DSC without a blocking layer, by contrast, the τn values obtained by the two methods do not agree and β cannot be determined.

Figure 8. Plots of τn vs light intensity for cells with and without blocking layers (IMVS data). The effect of the blocking layer is less evident here than in the plots of τn vs. photovoltage. The reason for this is explained in the text. The slope of the plot for the DSC with a blocking layer gives a lower value of β ) 0.11 (see text for explanation).

the quasi-stationary condition no longer holds when the photovoltage decay occurs predominantly via the substrate. In our previous work,6 we used IMVS to investigate the intensity dependence of τn to derive the electron diffusion length. It is interesting to note that it is not easy to distinguish between cells with and without blocking layers from a double logarithmic plot of τn vs photon flux. As Figure 8 shows, the slopes in both cases are quite similar. The cell with the blocking layer gives a slope of 0.89 in the double logarithmic plot, which corresponds to β ) 0.11 (compare simulated plots in Figure 4). This is lower that the value of β ) 0.19 obtained from the semilogarithmic plot of τn vs Uphoto. This difference expected, because it related to the fact that the intensity dependence of the photovoltage is not ideal (59 mV/decade), even for the cell with a blocking layer. The experimental intensity dependence of the photovoltage (70 mV/decade) may be a consequence of neglecting back-

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Figure 9. Density of states function st(E) derived from the photovoltage decay for the cell with a blocking layer. The DOS function is exponential with β ) 0.19 (cf. eq 2). The inset shows the semilogarithmic plot of st(E). The total density of active electron traps is estimated to correspond to approximately one per particle (see text for assumptions).

reaction via surface states.7 Alternatively, it may point to some unpinning of the conduction band position under illumination or to a small additional loss via recombination of electrons with the oxidized dye.7 At first sight it is difficult to see why τn is higher in the case of cells without a blocking layer. However, the reason is simple. These cells give a much lower photovoltage for a given photon flux, so that the quasi-Fermi level is much lower and τn is correspondingly smaller. It is therefore more useful to plot τn as a function of photovoltage (Figure 7), where the difference between the cells with and without blocking layer is immediately apparent. Because st(E) can only be derived with confidence if backreaction is excluded, analysis was restricted to cells with good blocking layers. Because the value of τn,0 is not known a priori, the absolute values of the trap DOS function are not accessible. For this, the charge extraction technique must be used.18 Nevertheless, the analysis using an assumed value of τn,0 is instructive because a semilogarithmic plot gives the value of β. A value of τn,0 ) 10-6 s was taken for the calculations shown here. Figure 9 shows that the trap density of states function is indeed exponential with β ) 0.19. Although the analysis of the cell without a blocking layer is not shown here, it is worth remarking that it gives an apparent value of β that is considerably higher than the true value. As pointed out above, the total trap density in the range Ef,dark to Ec cannot be determined unless τn,0 is known, a calculation based on the values assumed in the simulation (see Appendix) gives a value corresponding to approximately one electron trap per nanocrystallite for 30 nm particles. Work is in progress to compare this calculated value with the trapped electron density measured by a charge extraction technique.18 Conclusions The present work has demonstrated that it is essential to consider the influence of back-reaction at the conducting glass substrate when the transient and periodic photovoltage response

7398 J. Phys. Chem. B, Vol. 109, No. 15, 2005 of dye sensitized solar cells are analyzed. Unless steps are taken to block the substrate|electrolyte interface, the photovoltage decay will be influenced by the back-reaction, so that attempts to derive the relaxation time constantτn and the density of trap states function st(E) using the quasi-static approximation will lead to incorrect results. The analysis of data for cells with a compact blocking layer of TiO2 shows that it is possible to obtain reliable data. Further work is in progress to evaluate the influence of the blocking layer in other types of DSC, including those using organic hole conducting media. Acknowledgment. This work was supported by EPSRC and Johnson Matthey. We acknowledge helpful discussions with Alison Walker (Bath), as well as with Rainer Kern and Sarmimala Hore (Freiburg). Technical support was provided by Dr. Michael Bailes. We thank Professor Michael Gra¨tzel and Dr Jessica Kru¨ger for allowing P.J.C. to gain experience in the preparation of blocking layers by spray pyrolysis at EPFL. Appendix The following values were used in the simulations. Nc ) 1021 cm-3. Ec - EF,redox ) 1.1 eV. τn,0) 10-6 s. d ) 10-3 cm. Rn ) 0.5. Csub ) 0 F cm-2. st(0) ) 5 × 1015 cm-3. β ) 0.25. Initial photovoltage 0.7 V. References and Notes (1) O’Regan, B.; Gra¨tzel, M. Nature 1991, 353, 737. (2) Nelson, J. Phys. ReV. B-Condens. Matter 1999, 59, 15374.

Cameron and Peter (3) Cass, M. J.; Qiu, F. L.; Walker, A. B.; Fisher, A. C.; Peter, L. M. J. Phys. Chem. B 2003, 107, 113. (4) Fisher, A. C.; Peter, L. M.; Ponomarev, E. A.; Walker, A. B.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 949. (5) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. J. Phys. Chem. B 1997, 101, 10281. (6) Fisher, A. C.; Peter, L. M.; Ponomarev, E. A.; Walker, A. B.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 949. (7) Cameron, P. J.; Peter, L. M. J. Phys. Chem. B 2005, 109, 930. (8) Cameron, P. J.; Peter, L. M. J. Phys. Chem. B 2003, 107, 14394. (9) Duffy, N. W.; Peter, L. M.; Rajapakse, R. M. G.; Wijayantha, K. G. U. Electrochem. Commun. 2000, 2, 658. (10) Peter, L. M.; Duffy, N. W.; Wang, R. L.; Wijayantha, K. G. U. J. Electroanal. Chem. 2002, 524, 127. (11) Zaban, A.; Greenshtein, M.; Bisquert, J. ChemPhysChem 2003, 4, 859. (12) Bisquert, J.; Vikhrenko, V. S. J. Phys. Chem. B 2004, 108, 2313. (13) Peter, L. M.; Walker, A. B. Manuscript in preparation, 2004. (14) Schlichtho¨rl, G.; Huang, S. Y.; Sprague, J.; Frank, A. J. J. Phys. Chem. B 1997, 101, 8139. (15) Bisquert, J.; Zaban, A.; Greenshtein, M.; Mora-Sero, I. J. Am. Chem. Soc. 2004, 126, 13550. (16) Peter, L. M.; Wijayantha, K. G. U. Electrochim. Acta 2000, 45, 4543. (17) Kavan, L.; Gra¨tzel, M. Electrochim. Acta 1995, 40, 643. (18) Cameron, P. J.; Peter, L. M. Manuscript in preparation. (19) Zhu, K.; Schiff, E. A.; Park, N.-G.; van de Lagemaat, J.; Frank, A. J. Appl. Phys. Lett. 2002, 80, 685. (20) Franco, G.; Gehring, J,; Peter, L. M.; Ponomarev, E. A.; Uhlendorf, J. J. Phys. Chem. B 1999, 103, 692.